3. Philosophical Reflections On Mathematical Science
Cours de philosophie positive. (1/6) — Résumé 🇺🇸 English
This third lesson explores the philosophical foundations and overall system of mathematical science within positive philosophy. Although mathematics is the oldest and most advanced of all sciences, its general definition and primary divisions have historically remained vague and poorly determined. Traditional descriptions often label mathematics simply as the science of magnitudes or measurement. However, immediate and direct measurement of quantities is frequently impossible in practice, whether dealing with immense celestial distances, inaccessible terrain, or even basic vertical lines. This general inability to directly measure most desired quantities necessitates the creation of mathematical science.
Rather than relying on direct comparison, the human intellect determines quantities indirectly by connecting unknown magnitudes to directly measurable ones through precise relationships. Mathematics as a whole is therefore defined as the indirect measurement of magnitudes, where quantities are determined from one another according to the exact relations existing between them. This approach elevates mathematics from a mere collection of mechanical procedures into a vast chain of intellectual operations. By exemplifying how phenomena like falling bodies or geometric distances are analyzed through related variables, mathematics reveals its fundamental nature as the science of coordinating facts to deduce numerous results from minimal immediate data.
Every complete mathematical solution breaks down into two distinct parts that dictate the fundamental division of the science. The first part involves discovering the exact relations or equations governing the phenomena, constituting the concrete and experimental portion of the solution. The second part involves resolving the resulting numerical values through pure deduction, forming the abstract and rational portion of the science.
Mathematical analysis represents the most abstract, general, and fundamental science within positive philosophy, serving as the ultimate rational basis for all positive knowledge. Although historical mathematical conceptions often arose under the influence of geometry and mechanics, logical analysis remains entirely independent of them while providing the necessary foundation upon which they rest. This extreme simplicity of ideas accounts for the relative perfection of mathematical calculation compared to other sciences, transcending the mere utility of algebraic signs. Logically, mathematics possesses a rigorous universal character because every phenomenon can theoretically be conceived as a relation determining quantities through numbers.
Despite this absolute logical universality, practical limitations significantly restrict the effective domain of mathematical analysis as phenomena increase in complexity. Inorganic physics represents the maximum limit for successfully applying mathematical laws, as simpler inorganic bodies exhibit remarkable numerical fixity in their physical properties. Conversely, organic physiology and social phenomena display extreme numerical instability and immense irregular variations that fundamentally preclude precise mathematical treatment. Even within inorganic phenomena, complex conditions such as those found in meteorology or fluid dynamics create computational difficulties that surpass human capability, making celestial mechanics a rare exception due to particularly favorable planetary configurations.
Conscious of these vital boundaries, scientific inquiry must avoid the chimera of applying mathematics to realms governed by immense variability, ensuring each fundamental science retains its appropriate methodology. Building upon this overarching philosophical character of mathematics, the study now transitions to examining the three primary constituent sciences of calculus, geometry, and mechanics. The upcoming fourth lesson begins this specialized exploration by investigating mathematical analysis and deeply examining the fundamental nature of equations.
Essential Elements Of Mathematical Analysis
This third part examines the foundational elements that constitute all analytical combinations within mathematics. Despite an infinite variety of potential functions, calculations ultimately rely on a very small, well-defined set of elementary functions of a single variable, paired with their inverse functions. These simple abstract functions, alongside complementary concepts like circular and elliptic functions, provide the necessary framework for determining the abstract or concrete nature of mathematical problems. However, because this inventory of fundamental analytical elements remains limited, establishing direct equations between real-world phenomena and abstract numbers presents an immense, fundamental difficulty for the human intellect.
To overcome this barrier, mathematics divides its computational processes into two distinct branches, namely algebraic calculation and arithmetic calculation. Algebraic calculation, or algebra, focuses on transforming equations to reveal the exact relationships and modes of formation between unknown and known quantities, essentially functioning as a calculus of functions. Arithmetic calculation, or arithmetic, subsequently deals with evaluating these formulas to determine the precise numerical values of the quantities, acting as a calculus of values. Although arithmetic encompasses specialized fields like the theory of numbers, its actual domain is infinitely narrower than that of algebra, as evaluation simply translates numbers into regular numerical forms through transformations.
Recognizing the strict limitations in creating brand new elementary functions, the human mind developed a powerful auxiliary strategy known as transcendent analysis to simplify the establishment of equations. Instead of dealing directly with primary magnitudes, this advanced approach introduces auxiliary quantities such as infinitesimals, fluxions, or derivatives, allowing mathematicians to navigate complex physical phenomena. Whether viewed through the conceptual lenses of Leibniz, Newton, or Lagrange, these transcendent methods provide the essential analytical machinery needed to expand the reach of mathematical science while maintaining rigorous logical foundations.
The resolution of equations forms the ultimate goal of functional calculation, dividing naturally into algebraic and numerical branches based on the nature of analytical elements. While algebraic equations are classified rationally by their degrees, the rigorous resolution of general algebraic equations has only been successfully achieved up to the fourth degree, as higher degrees surpass the effective limits of human intelligence. Consequently, analysts have increasingly relied on numerical resolution to determine specific values independently of general formulas, despite the cumbersome complications and iterative approximations involved.
This structural division highlights the distinct methodologies required for different mathematical problems. Algebraic methods focus purely on functions and relationships without regard to numerical values, whereas numerical methods must account for specific quantitative distinctions such as commensurable and incommensurable roots. Furthermore, underlying these techniques is the broader theory of equations, which utilizes methods like indeterminate coefficients and function transformation to expand analytical resources.
Specialized algebraic expressions, including imaginary numbers and negative quantities, often spark unnecessary metaphysical debates that obscure their true analytical function. When viewed purely as abstract relationships rather than fixed values, these anomalous expressions demonstrate the absolute consistency and logical simplicity of mathematical analysis. Ultimately, while abstract theories of negative quantities are complete, their concrete interpretations rely heavily on geometric and mechanical verification, reinforcing the vital connection between abstract formulation and physical reality.
The text concludes the general considerations regarding the calculation of direct functions while acknowledging the natural limitations of the course length. It then transitions directly to the upcoming philosophical examination of the calculation of indirect functions. This subsequent branch of mathematics possesses a significantly greater importance and broader scope, thereby demanding a more extensive and detailed development. A concluding note reflects upon the author's development of a general theory of homogeneity twelve years prior through daily teaching. It mentions a parallel discovery made by M Fourier in his prominent 1822 work on heat, noting that this natural convergence arose from investigating a simple subject independently of any special application.